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Abstract This paper is devoted to the phenomenon of superintegrability. This phenomenon is manifested in the existence of a formula for character averages, expressed through the same characters at special points and of its various generalization. In this paper we develop a method of proving such formulas from first principle from Virasoro constraints and W -representation. We apply it to prove the formula for the Jack functions averages – appropriate analogue of characters for the β -deformed Hermitian Gaussian matrix model. We also sketch the construction of W -operators from Calogero–Ruijsenaars Hamiltonians.
Mishnyakov et al. (Wed,) studied this question.